Binary decision making with very heterogeneous influence

Let us consider the following simple problem. Suppose we have N independently and identically distributed (iid) random variables Jⁱ (i = 1,…,N) distributed according to the distribution

$$\begin{eqnarray} &&P({J}^{i}\gt x)=\max (1,x)^{-\nu } \end{eqnarray} \tag{ 12 }$$

for 0 < ν < 1. For the remainder of this section, we will assume that x ≥ 1, to simplify notation. Without loss of generality, we can re-label these variables as J_i, with the re-labeling chosen so that J₁ > J₂ > ⋯ > J_N.¹ The cumulative distribution function for J_k is easy to find:

$$\begin{eqnarray} &&\fl P({J}_{k}\lt x)=\left (\begin{array}{@{}l@{}} \displaystyle N\\ \displaystyle k-1 \end{array}\right )\left ({x}^{-\nu }\right )^{k-1}\left (1-{x}^{-\nu }\right )^{N-k+1}\nonumber\\ &&+~\left (\begin{array}{@{}l@{}} \displaystyle N\\ \displaystyle k-2 \end{array}\right )\left ({x}^{-\nu }\right )^{k-2}\left (1-{x}^{-\nu }\right )^{N-k+2}+\cdots \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} &&=~\left (1-{x}^{-\nu }\right )^{N}\sum _{j=0}^{k-1}\left (\begin{array}{@{}l@{}} \displaystyle N\\ \displaystyle j \end{array}\right )\left (\frac{1}{{x}^{\nu }-1}\right )^{j}. \end{eqnarray} \tag{ 14 }$$

The interpretation is simple: the first term corresponds to the probability that J_k−1 are all bigger than x (the multiplicative factor corresponds to the number of possible permutations of J_k−1 from the original J^α). However, there is also the probability that J_k−1 is also smaller than x, etc, and accounting for all of these possibilities we obtain the total sum. When N is large and x ≫ 1, we can make the approximation

$$\begin{eqnarray} &&P({J}_{k}\lt x)\approx \exp \left [-\frac{N}{{x}^{\nu }}\right ]\sum _{j=0}^{k-1}\left (\begin{array}{@{}l@{}} \displaystyle N\\ \displaystyle j \end{array}\right ){x}^{-j \nu }. \end{eqnarray} \tag{ 15 }$$

From equation (15) we see that the scale for J to be large is N^1/ν: in particular, the probability distribution on J₁ is rapidly suppressed (as an exponential in N) for x ≪ N^1/ν. The probability that J₁ < N^1/ν is 1 − 1/e. The probability that there are exactly k Jⁱs which are larger than N^1/ν is given by

$$\begin{eqnarray} &&P(k\;\mathrm{large}\;J\mathrm{s})=P({J}_{k+1}\lt {N}^{1/\nu })-P({J}_{k}\lt {N}^{1/\nu })\approx \frac{1}{e k !}. \end{eqnarray} \tag{ 16 }$$

It is straightforward to argue that the scale of the typical J_i < N^1/ν is

$$\begin{eqnarray} &&\left \langle \sum _{i=1}^{N}{J}^{i}\Theta \left ({N}^{1/\nu }-{J}^{i}\right )\right \rangle \approx N\int \nolimits _{1}^{{N}^{1/\nu }}\mathrm{d}x \hspace{0.167em} \frac{\nu x}{{x}^{\nu +1}}\approx \frac{\nu }{1-\nu }{N}^{1/\nu } \end{eqnarray} \tag{ 17 }$$

with standard deviation

$$\begin{eqnarray} &&\left [\mathrm{Var}\left (\sum _{i=1}^{N}{J}^{i}\Theta \left ({N}^{1/\nu }-{J}^{i}\right )\right )\right ]^{1/2}\approx \sqrt{\frac{\nu }{2-\nu }}{N}^{1/\nu }. \end{eqnarray} \tag{ 18 }$$

These manipulations suggest that only a few influence variables dominate the sum J₁ + ⋯ + J_N. It therefore seems reasonable to focus only on the largest few random variables in this sum. In particular, we will focus on the random variables

$$\begin{eqnarray} &&{K}_{i}=\frac{{J}_{i}}{{J}_{1}+\cdots +{J}_{N}} \end{eqnarray} \tag{ 19 }$$

in this section, as these are the weights that enter into equation (8) for Q in the binary decision model on complete graphs. Of course, it really is no different on any other graph either, as although the denominator only contains k_v + 1 terms for node v, the sum will still be dominated by only a few terms. More importantly, while J_i have divergent and badly behaved distributions, the distributions on K_i are well-behaved. In general, it is very difficult to compute the distributions of K_i. However, we notice that if ν is close to 0, equations (17) and (18) suggest that all of the small J added together are still significantly smaller than the largest few J.

Let us begin by looking at the distribution on the ratios J_k/J₁. The cumulative joint distribution on J_k and J₁ is given by

$$\begin{eqnarray} &&P({J}_{k}\lt y,{J}_{1}\lt x)=\left (1-{y}^{-\alpha }\right )^{N}\sum _{j=0}^{k-1}\left (\begin{array}{@{}l@{}} \displaystyle N\\ \displaystyle j \end{array}\right )\left (\frac{{y}^{-\alpha }-{x}^{-\alpha }}{1-{y}^{-\alpha }}\right )^{j}. \end{eqnarray} \tag{ 20 }$$

To very good approximation when N is large, we have

$$\begin{eqnarray} &&\fl P\left (\frac{{J}_{k}}{{J}_{1}}\gt z\right )=\int \nolimits _{1}^{\infty }\mathrm{d}y\frac{\partial }{\partial y}P({J}_{k}\lt y,{J}_{1}\lt x)\biggl \vert _{x=y/z} \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} &&\approx ~\int \nolimits _{1}^{\infty }\mathrm{d}y \hspace{0.167em} \frac{\nu }{{y}^{\nu +1}}{\mathrm{e}}^{-N{y}^{-\nu }}\nonumber\\ &&\times ~\left [N\sum _{j=0}^{k-1}\frac{{N}^{j}}{j !}\left ({y}^{-\nu }-{y}^{-\nu }{z}^{\nu }\right )^{j}-\sum _{j=1}^{k-1}\frac{{N}^{j}}{(j-1)!}\left ({y}^{-\nu }-{y}^{-\nu }{z}^{\nu }\right )^{j-1}\right ] \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} &&\approx ~\int \nolimits _{1}^{\infty }\mathrm{d}y \hspace{0.167em} \frac{\alpha N}{{y}^{\nu +1}}{\mathrm{e}}^{-N{y}^{-\nu }}\frac{{N}^{k-1}}{(k-1)!}{y}^{-\alpha (k-1)}\left (1-{z}^{\nu }\right )^{k-1}\nonumber\\ &&=~\int \nolimits _{0}^{N}\mathrm{d}u \hspace{0.167em} {\mathrm{e}}^{-u}\frac{{u}^{k}}{(k-1)!}\left (1-{z}^{\nu }\right )^{k-1} \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} &&\approx ~\left (1-{z}^{\nu }\right )^{k-1}. \end{eqnarray} \tag{ 24 }$$

From here we can compute

$$\begin{eqnarray} &&\left \langle \frac{{J}_{k}}{{J}_{1}}\right \rangle =\int \nolimits _{0}^{1}\mathrm{d}\left (1-{z}^{\nu }\right )^{k-1}z=\frac{\Gamma (1+1/\nu )\Gamma (k)}{\Gamma (k+1/\nu )}. \end{eqnarray} \tag{ 25 }$$

This asymptotes to

$$\begin{eqnarray} &&\left \langle \frac{{J}_{k}}{{J}_{1}}\right \rangle \sim \frac{1}{{k}^{1/\nu }}. \end{eqnarray} \tag{ 26 }$$

Now, let us study what happens in the large N limit. In particular, we want to understand the distribution on K₁. As we will confirm in the next sections, whenever the probability that K₁ > 0 is finite, even as N → ∞, the mean field description of the binary decision model breaks down. This is because in this case, the macroscopic dynamics of the model will depend on the microscopic details: in particular, the realization of disorder P₁ for the most influential node. Therefore, in this section, we will determine P(K₁ > 0) (after the N → ∞ limit).

Let us begin with the case 0 < ν < 1. In this case, we can prove that P(K₁ > 0) = 1 by showing that 〈1/K₁〉 is finite in the limit N → ∞:

$$\begin{eqnarray} &&\left \langle \frac{1}{{K}_{1}}\right \rangle =\sum _{k=1}^{N}\left \langle \frac{{J}_{k}}{{J}_{1}}\right \rangle \sim \sum _{k=1}^{N}\frac{1}{{k}^{1/\nu }}\lt \infty . \end{eqnarray} \tag{ 27 }$$

We thus see that almost surely there is no mean field limit in this case.

Next, let us look at the case ν = 1. In this case, we see that J₁ is almost surely of order N.² The sum of all of the smaller random variables is of the order

$$\begin{eqnarray} &&\left \langle \sum _{i=2}^{N}{J}_{i}\Theta ({J}_{1}-{J}_{i})\right \rangle \sim N\int \nolimits _{0}^{{J}_{1}}\frac{\mathrm{d}{J}_{1}}{{J}_{1}}\sim N\log {J}_{1}\sim N\log N \end{eqnarray} \tag{ 28 }$$

with standard deviation

$$\begin{eqnarray} &&\left [\mathrm{Var}\left (\sum _{i=2}^{N}{J}_{i}\Theta ({J}_{1}-{J}_{i})\right )\right ]^{1/2}\sim (N{J}_{1})^{1/2}\sim N. \end{eqnarray} \tag{ 29 }$$

Thus in the strict limit N → ∞ we see that there is a mean field limit almost surely. However, this is a very slow limiting process, and we will return to this point shortly.

The case ν > 1 also has a mean field limit, as it should. To explicitly check this, we see that N^1/ν ∼ J₁ ≪ J_tot ∼ N, and

$$\begin{eqnarray} \fl \mathrm{Var}({J}_{\mathrm{tot}})^{1/2}\sim {N}^{1/2}\left [\int \nolimits _{1}^{N}\mathrm{d}{J}_{1}\hspace{0.167em} {J}_{1}^{1-\nu }\right ]^{1/2}\sim \left \{ \begin{array}{@{}ll@{}} \displaystyle {N}^{1/\nu }&\displaystyle \tqs 1\lt \nu \lt 2\\ \displaystyle \sqrt{N\log N}&\displaystyle \tqs \nu =2\\ \displaystyle \sqrt{N}&\displaystyle \tqs \nu \gt 2. \end{array}\right . \end{eqnarray} \tag{ 30 }$$

When ν > 2, of course, the standard central limit theorem holds and J_tot is essentially a Gaussian distribution. Regardless, we see that for ν > 1, fluctuations are subleading just as we expect, so the mean field limit exists.

We should also note that even if there is formally a mean field limit, this mean field limit may require N to be so large that it is impractical for ‘realistic’ situations. To that end, let us alternatively say that the mean field limit occurs when there are almost surely (away from any possible phase transitions) no avalanches of size larger than λ. In any linear response regime, we have shocks no larger than Δq ∼ K₁/(1 − α), which simply follows from the fact that having many influential nodes all having very similar P values is very unlikely. Note that in the case where there is no influence, K₁ = 1/N, and the total number of nodes that flips on average reduces to 1/(1 − α), as was discussed in section 2. We conclude that when K₁ ≪ λ, we have a mean field limit, and when K₁ ≫ λ, we do not³.

Now, we have already seen that K₁ is almost surely positive when ν < 1, so there can be no mean field limit here, so long as λ is chosen to be small enough. In the case ν = 1, we have K₁ ∼ 1/logN, which means that if we are looking at scale λ, the size of N required to have a mean field limit is N ∼ exp[1/λ], which is likely much larger than any practical graph for any reasonable resolution λ. When ν > 1, we require N^1/ν−1 ≪ λ, or N ≫ (1/λ)^ν/(ν−1), which may be ‘surprisingly’ large. For example, in the case ν = 2 which is believed to be experimentally relevant in many systems, we actually need to have N ∼ 1/λ² to have a mean field limit, and if our resolution λ is small, this limit may be beyond our reach with any practical social system. We should finally note that the limit ν → ∞ results in N ∼ 1/λ, which is trivial since the number of nodes which flip in any avalanche must be an integer multiple of 1/N. This final case where N ∼ 1/λ will hold, up to minor corrections, in most systems where J₁ does not have a power law tail distribution: for example, in the case of a Gaussian distribution on the J, we have ${K}_{1}\sim {N}^{-1}\sqrt{\log N}$, which is not much different from N⁻¹.

As we demonstrated in section 4, there can be no mean field limit in the case where ν < 1. The consequences of the mean field limit being non-existent are dramatic: the macroscopic dynamics of the system are sensitive to the realization of P_v, which is microscopic data. This should make sense, since when ν < 1, a finite number of nodes can carry a finite fraction of the total influence in the graph in the limit N → ∞. So we begin by showing in figure 1 some typical trajectories of q(p), found by increasing p starting from q(0) = 1. Since we are working on a complete graph, we can write very simple code by using the fact that we can order the nodes in order of their values of P_v, and sequentially look through them to determine whether or not they will be in state 0 or 1 at a given value of p. Exploiting the ‘graph structure’ here allows us to work with orders of magnitude more nodes with fast run times. Note that despite N ranging over four orders of magnitude, there is no noticeable mean field limit, just as we predicted earlier. Also note, particularly on the uniform distribution, the possibility of multiple large shocks.

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There have been previous models which admit shocks without critical phenomena—admitting lower cutoffs in the distribution of P_v can result in sudden shocks, as was demonstrated in [22, 26]. However, the key difference with these models is that in the former models, the location of this shock is predictable given the microscopic dynamics. Furthermore, the shock in these models is typically of the form q(p) = Θ(p_c − p), which means that the entire market changes its state in one global avalanche. Although this possibility should not be ruled out, since some economic markets (e.g., housing) will never undergo such radical binary shifts, alternative mechanisms for unpredictability should be explored. One mechanism which is slightly simpler mathematically than heavy-tailed influence distributions is to allow the function F(P) distribution to have discontinuous first derivative (or even be discontinuous) at points where 0 < F(P) < 1. However, this mechanism seems rather non-robust and would require the model to be fine-tuned.

Now, let us return to unpredictable decision making, and focus on some analytic computations which help illuminate the underlying behavior of these systems. The picture gallery makes clear that the various phases of this model, encapsulated in the double-valued q(p), is itself a random variable in unpredictable decision making. However, this does not imply that the model cannot be solved in any limit; instead, what we mean by ‘solvability’ must be slightly re-defined. This is precisely analogous to the way in which the fair coin flip problem is exactly solved—there is an equal probability for the coin to give heads or tails, although on a given realization the result will be unknown. Similarly, for this problem, the easily solvable limit corresponds to ν → 0, in which only one node should dominate the overall influence, and therefore the macroscopic behavior. Higher order corrections will become more and more relevant for larger ν, in which the behavior of two, three, etc nodes must be accounted for.

In this limit, it is very easy to understand the model, to first order. The first order approximation for this computation is that exactly one node contributes to Q: i.e. 

$$\begin{eqnarray} &&Q={x}_{1}. \end{eqnarray} \tag{ 31 }$$

Here x₁ refers to the state of the most influential node. The decision making will be bistable and will correspond to solutions of the pair of equations

$$\begin{eqnarray} &&{x}_{1}=\Theta ({P}_{1}h({x}_{1})-p), \end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray} &&q=F\left (\frac{p}{h({x}_{1})}\right ). \end{eqnarray} \tag{ 33 }$$

From these equations it is straightforward to derive that q(p) is a multivalued function with upper branch q₊(p) and lower branch q₋(p):

$$\begin{eqnarray} {q}_{+}(p)=\left \{ \begin{array}{@{}ll@{}} \displaystyle F(p/h(1))&\displaystyle \tqs {P}_{1}h(1)\lt p\\ \displaystyle F(p)&\displaystyle \tqs {P}_{1}h(1)\gt p, \end{array}\right . \end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray} {q}_{-}(p)=\left \{ \begin{array}{@{}ll@{}} \displaystyle F(p/h(1))&\displaystyle \tqs {P}_{1}\lt p\\ \displaystyle F(p)&\displaystyle \tqs {P}_{1}\gt p, \end{array}\right . \end{eqnarray} \tag{ 35 }$$

insofar as the first order approximation that only one node has influence is relevant. The locations of the discontinuity on q₊ and on q₋, as well as the size of the discontinuity, is determined by the random variable P₁, the intrinsic value (internal magnetic field) as determined by the most influential node in the graph. Nonetheless, P₁ is microscopic data about the state of the system, so to an outside observer who cannot determine this, the behavior will appear unpredictable.

In this section, our main numerical result will be to compute the probability density function of s₁, which is the size of the shock during which the most influential node flips from 1 to 0. The size of the shock is easily shown to be

$$\begin{eqnarray} &&{s}_{1}=F({P}_{1})-F(h(1){P}_{1}) \end{eqnarray} \tag{ 36 }$$

and therefore the distribution on s₁ can be found straightforwardly by studying the preimage of the function of the right-hand side.

In general it is quite ugly to invert equation (36). In the case where P₁ is distributed uniformly on [0,1] and h = 1 + AQ, however, it is straightforward to compute

$$\begin{eqnarray} {s}_{1}=\left \{ \begin{array}{@{}ll@{}} \displaystyle A{P}_{1}&\displaystyle \tqs {P}_{1}\lt 1/(A+1)\\ \displaystyle 1-{P}_{1}&\displaystyle \tqs {P}_{1}\geq 1/(A+1). \end{array}\right . \end{eqnarray} \tag{ 37 }$$

This implies that s₁ is uniformly distributed on [0,A/(A + 1)].

From the picture gallery, it is clear that in general a first order approximation is not enough. This is because there are multiple jumps (shocks) in many of the curves (note that the number of large jumps is random as well), and whenever there are two or more shocks, that implies that at least two influential nodes were relevant.⁴ We will see that numerical data demonstrates this is true as well for the distribution of s₁. Let us therefore also describe the second order approximation, which already becomes quite cumbersome. We can tackle this most straightforwardly by first breaking the problem into two problems, each of which happens with half probability: P₁ < P₂ and P₂ < P₁. Since K₁ = 1 − K₂, P₁ and P₂ are each independent, we will assume that K₁ is fixed and average over its distribution at the very end. To begin, let us denote P₋ = min(P₁,P₂) and P₊ = max(P₁,P₂). We can write the probability density function on the shock size s₁ by computing the contributions from the probability that P₁ < P₂ and both nodes are in different shocks, then the same shock; and repeating for P₂ < P₁:

$$\begin{eqnarray} &&\fl p(s\vert {K}_{1})=\int _{{P}_{-}\lt {P}_{+}}(-{F}^{\prime}({P}_{-})\hspace{0.167em} \mathrm{d}{P}_{-})(-{F}^{\prime}({P}_{-})\hspace{0.167em} \mathrm{d}{P}_{+})\nonumber\\ &&\times ~\left [\delta \left (s-F({P}_{-})+F\left (\frac{h(1)}{h({K}_{2})}{P}_{-}\right )\right )\Theta \left ({P}_{+}-\frac{h(1)}{h({K}_{2})}{P}_{-}\right )\right .\nonumber\\ &&+~\delta \left (s-F({P}_{-})+F\left (h(1){P}_{-}\right )\right )\left (\Theta \left (\frac{h(1)}{h({K}_{2})}{P}_{-}-{P}_{+}\right )\right .\nonumber\\ &&+~\left .\Theta \left (\frac{h(1)}{h({K}_{1})}{P}_{-}-{P}_{+}\right )\right )\nonumber\\ &&\left .+~\delta \left (s-F\left (\frac{h(1)}{h({K}_{1})}{P}_{+}\right )+F(h(1){P}_{+})\right )\Theta \left ({P}_{+}-\frac{h(1)}{h({K}_{1})}{P}_{-}\right )\right ]. \end{eqnarray} \tag{ 38 }$$

In the case of P₁ and P₂ uniformly distributed on [0,1] with h(Q) = 1 + AQ, this integral can be explicitly evaluated:

$$\begin{eqnarray} &&\fl p(s\vert {K}_{1})=\Theta (s)\Theta \left (\frac{A{K}_{1}}{1+A}-s\right )\left (\frac{1+A(1-{K}_{1})}{A{K}_{1}}+s\left (1-\frac{(1+A)(1+A(1-{K}_{1}))}{(A{K}_{1})^{2}}\right )\right )\nonumber\\ &&\nonumber\\ &&+~\Theta (s)\Theta \left (\frac{A}{A+1}-s\right )\left (\frac{1-{K}_{1}}{1+A{K}_{1}}+\frac{{K}_{1}}{1+A(1-{K}_{1})}\right )\frac{s}{A}\nonumber\\ &&+~\Theta \left (\frac{A}{A+1}-s\right )\Theta \left (s-\frac{A{K}_{1}}{1+A}\right )\frac{A{K}_{1}(1-s)}{1+A(1-{K}_{1})}\nonumber\\ &&+~\Theta (s)\Theta \left (\frac{A(1-{K}_{1})}{1+A}-s\right )s\nonumber\\ &&+~\Theta \left (s-\frac{A(1-{K}_{1})}{1+A}\right )\Theta \left (\frac{A}{A+1}-s\right )\frac{A(1-{K}_{1})(1-s)}{1+A{K}_{1}}\nonumber\\ &&+~\Theta (s)\Theta \left (\frac{A{K}_{1}}{1+A{K}_{1}}-s\right )\left (\frac{1+A{K}_{1}}{1+A}\right )^{3}\left (1+s\left (\frac{1}{(A{K}_{1})^{2}}-1\right )\right ). \end{eqnarray} \tag{ 39 }$$

At this point, we simply perform the average over K₁ numerically, by using equation (24). Figure 2 shows that for a small value of ν, there is excellent agreement between the theoretical prediction and numerical data, suggesting that in this limit, only the first two nodes are relevant. Note that the first order approximation would not have sufficed.

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Glassy behavior comes about from the possibility that a cluster of nodes could simultaneously be all in state 1 or all in state 0, as discussed in section 2. However, this picture breaks down in the very heterogeneous influence model, because most nodes are only following a single node. This must remain true on random graphs, because the Lévy flights associated with the total influence that each node sees are dominated by a few terms regardless of the number of terms in the sum.

Let us assume that the random graph is sparse in that the degree of any node is much smaller than the total number of nodes. Let us also assume that ν is close to 0—we will return to what happens at larger values of ν later. Although the most influential node cannot influence the entire graph as it can on the complete graph, it does still entirely determine the state of each of its neighbors. Now, let us imagine removing the influential node and all of its neighbors from the graph. We can perform the same procedure again, and remove more nodes which are mostly entirely following a different node. Continuing this process until the graph has no remaining nodes, we see that the graph decomposes into local clusters which are effectively described by the unpredictable model on a complete graph. Since we have many of these clusters, each of which have independent thresholds, we expect a well-behaved and predictable thermodynamic limit to emerge.

Let us be more quantitative about this resulting limit. To do this, let us first focus on a single cluster. Within a single cluster, the state is entirely determined by P₁, the internal field of the most influential node. Each cluster is characterized by a configuration with a regime of bistability at intermediate p, which can be described by specifying the upper and lower branches of the multivalued function q(p), which is precisely the result we found earlier in equation (35). Now, let us average over the value of P₁. This average is actually quite simple to take:

$$\begin{eqnarray} &&{q}_{+}(p)=F\left (\frac{p}{h(1)}\right )^{2}+\left [1-F\left (\frac{p}{h(1)}\right )\right ]F(p), \end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray} &&{q}_{-}(p)=F(p)F\left (\frac{p}{h(1)}\right )+\left [1-F(p)\right ]F(p). \end{eqnarray} \tag{ 41 }$$

So we conclude that if F(p) is a continuous function, this model is in fact characterized by a single, glassy phase with equilibrium spectrum width

$$\begin{eqnarray} &&\Delta q(p)\equiv {q}_{+}(p)-{q}_{-}(p)=\left [F\left (\frac{p}{h(1)}\right )-F(p)\right ]^{2}. \end{eqnarray} \tag{ 42 }$$

We emphasize that this glassy phase not only is independent of N, as it must be to be a well-behaved thermodynamic limit, but it is entirely independent of the graph, so long as the degree of each node is much smaller than N.

Figure 3 demonstrates that qualitatively, the theory is very accurate at small values of ν, and becomes less so at larger values of ν. We will return to this point in a later section. Note that the glassy phase described above is remarkably independent of the underlying network structure, just as we predicted. Figure 4 shows typical samples of the total glassy phase at finite values of 〈k〉, when compared to the approximately 〈k〉-independent result. One can see that the larger the value of 〈k〉, the larger the typical shocks, since the clusters which are flipping contain more nodes.

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At a finite value of N, there will of course be fluctuations in mean field theory. For each cluster, the variance is given by

$$\begin{eqnarray} &&\fl \mathrm{Var}({q}_{+}(p))_{1}=F\left (\frac{p}{h(1)}\right )F\left (\frac{p}{h(1)}\right )^{2}+\left [1-F\left (\frac{p}{h(1)}\right )\right ]F(p)^{2}\nonumber\\ &&-~\left [F\left (\frac{p}{h(1)}\right )^{2}+\left [1-F\left (\frac{p}{h(1)}\right )\right ]F(p)\right ]^{2} \end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray} &&=~F\left (\frac{p}{h(1)}\right )\left [1-F\left (\frac{p}{h(1)}\right )\right ]\left [F\left (\frac{p}{h(1)}\right )-F(p)\right ]^{2} \end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray} &&\fl \mathrm{Var}({q}_{-}(p))_{1}=F\left (p\right )F\left (\frac{p}{h(1)}\right )^{2}+\left [1-F\left (p\right )\right ]F(p)^{2}\nonumber\\ &&-~\left [F\left (p\right )F\left (\frac{p}{h(1)}\right )+\left [1-F\left (p\right )\right ]F(p)\right ]^{2} \end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray} &&=~F(p)(1-F(p))\left [F\left (\frac{p}{h(1)}\right )-F(p)\right ]^{2}. \end{eqnarray} \tag{ 46 }$$

For the total graph, we just need to divide by the number of clusters, which is of order N/〈k〉, as each cluster behaves independently, so we expect

$$\begin{eqnarray} &&\mathrm{Var}(q(p))\sim \frac{\langle k\rangle }{N}\mathrm{Var}(q(p))_{1}. \end{eqnarray} \tag{ 47 }$$

Surprisingly, for many regions of parameter space this relation is obeyed fairly quantitatively (to within a factor of 2), despite our crude estimate of the number of clusters, as shown in figure 5. The scaling in 〈k〉 is always obeyed although sometimes the effective prefactor appears to differ—depending on what the internal distribution F(P) is.

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Let us now compare the glassy phase of the very heterogeneous influence model with the other case—where influence is not ‘heavy-tailed’. Making precise what we mean by ‘heavy-tailed’ is likely quite cumbersome, and for this paper we will simply continue to refer to J distributions with ν < 1 as heavy-tailed.

To understand what happens when the influence distribution is not heavy-tailed, consider the opposite limiting case where J ≈ 1, with small amounts of disorder in influence. Influence in this limit has almost no effect on the equilibria. We have seen this already at mean field level, but it also holds to higher orders in the cavity expansion. We say ‘almost’ here because there is a very small effect, which is analogous to the effect of small degrees in the homogeneous influence model. To write the cavity equation, let us denote the state of the cavity node with x₀, and one of its neighbors with x₁. Let K₁₀ denote the relative influence of the cavity on its neighbor. Averaging over the neighbors of the cavity, we find analogously to the computation in [22] that the TAP equations become

$$\begin{eqnarray} &&P({x}_{1}=1)=q+\alpha \langle {K}_{1 0}\vert {J}_{0}\rangle ({x}_{0}-q). \end{eqnarray} \tag{ 48 }$$

The first term in this equation is the mean field result, and the second term is the cavity’s effect on its neighbors—for a more careful derivation, one should consult the reference above. From the x₀-dependent piece of this equation, we can extract that the probability that both a 0 and 1 are allowed solutions:

$$\begin{eqnarray} &&P({x}_{0}=0\;\mathrm{or}\;1)={\alpha }^{2}\langle {K}_{1 0}\rangle , \end{eqnarray} \tag{ 49 }$$

where we have averaged over P₀ and J₀, about the mean field solution. But 〈K₁₀〉 = 1/〈k〉, a fact which is easy to see by the following logic:

$$\begin{eqnarray} &&\langle \langle {K}_{1 0}\vert {J}_{0}\rangle _{{k}_{1}}\rangle _{{J}_{0}}=\langle \langle {K}_{1 0}\vert {k}_{1}\rangle _{{J}_{0}}\rangle _{{k}_{1}}=\langle {k}_{1}^{-1}\rangle _{{k}_{1}}=\frac{1}{\langle k\rangle }. \end{eqnarray} \tag{ 50 }$$

We have used the tower property of conditional expectation values here, and used that averaging over all influences, the influence of any one neighbor is equally likely to that of any other, and therefore the influence must be the inverse of the degree. We conclude, using this logic, that the first order cavity correction to the mean field equations is therefore independent of influence. Of course, we showed with a heuristic argument earlier that very heterogeneous influence models do alter the glassy physics, and confirmed this numerically. The flaw in this argument is simply that we cannot linearize around a solution in the very heterogeneous influence model, since the behavior of x₀ is entirely dominated by the most influential node.

So we conclude that for ν ≪ 1, we should have the highly glassy phase described in earlier sections, and for ν ≫ 1 we should return to the glassy phase of [22], given by equation (7), which has small 1/〈k〉 deviations from mean field theory (with the possible suppression of discontinuities in phase transitions). Figure 6 shows the transition between the two glassy phases, as ν is varied, for two values of 〈k〉.⁵ It is likely that for ν ∼ 1 there is a complicated interplay between finite degree effects in the network and multiple influential nodes being relevant, and so we do not attempt a further theoretical analysis of this case. We do make a final note, however: we see numerical evidence that influence may suppress a phase transition even when ν > 1. In the simulations with 〈k〉 = 40, there is a genuine phase transition at ν = 5; however, at ν = 1.5, we see a ‘tail’ in q(p). It is reasonable to suspect that a wide enough distribution in influence may thus have a benign effect on the global decision making process by helping to smooth out shocks and other unfortunate behavior.

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In many cases, such as models of the type presented in [33], it does not make sense to ignore correlations between degree and influence. In this section, we will consider what happens when we consider binary decision making with influence which translates into graph structure according to some relation k_v = k(J_v) with k(J) a monotonically increasing function. For this paper, we will focus on the simple function

$$\begin{eqnarray} &&k(J)={k}_{0}+{J}^{\theta } \end{eqnarray} \tag{ 51 }$$

with J distributed as usual according to equation (12) and 0 < θ some power. We will choose to construct our graphs as random graphs made with the Molloy–Reed algorithm [23, 24]. For large k, the degree distribution of the network will be given by

$$\begin{eqnarray} &&{\rho }_{k}\sim {k}^{-1-\nu /\theta }. \end{eqnarray} \tag{ 52 }$$

Depending on the values of ν and θ, we thus expect a variety of possible behaviors to occur.

If ν ≤ θ, we see that the distribution on ρ_k is heavy-tailed enough that some nodes in the graph should touch a finite fraction of the nodes. This implies that very influential nodes have many connections. If ν < 1, we also conclude that there is no mean field limit and therefore the decision making is unpredictable. If ν > 1, then unless 〈k〉 is very small, we expect that we quickly transition into a homogeneous-like glass phase. To see this, we note that all of the most influential nodes are connected to each node, and since for ν > 1 the very influential nodes will count roughly equally, the independence of P_v from J_v implies that each node will see a crude ‘mean field’ collection of nodes. If ν > θ, then the distribution on ρ_k is not heavy-tailed enough, and the most connected node will most likely not reach a finite fraction of nodes in the graph. This implies that there is a mean field limit. If ν < 1, the mean field limit will be qualitatively similar to the universal glassy phase described in section 5.3, and if ν > 1, the mean field limit will be the homogeneous influence limit.

Figure 7 shows that these qualitative arguments hold in practice in our simulations. As in section 5.3, ‘far’ from limiting cases these arguments do not hold perfectly, but it is clear what the basic mechanisms at play are. When ν < 1, the average q(p) is qualitatively very similar to the Erdös–Rényi case described in the previous sections, with minor quantitative differences due to θ > 0. As θ becomes larger than ν, the size of the shocks becomes increasingly large, although due to numerical limitations we could not observe a ‘sharp’ transition between behaviors at ν = θ. The ν > 1 case depends very little on the value of θ, and only on 〈k〉. In general, we see substantial robustness of the model to precise details of the graph structure, at least for the random graph ensembles we have studied.

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