Absolute value of the difference between the extinction time distribution <i>P</i>_<i>ext</i>(<i>t</i>) computed from direct sampling with complete information and those calculated with density sampling (blue), BP (red) and Similarity Sampling (magenta).

<p>a) On trees of <i>N</i> = 1092 nodes, with branching ratio 3 (〈<i>k</i>〉 ≈ 2) and with uniform epidemic parameters λ = 0.7, <i>μ</i> = 0.5. The partial observation is performed sampling uniformly the state of 10% of the nodes at <i>T</i>_<i>obs</i> = 5 and averaging over <i>M</i>_<i>o</i> = 210 such realizations. b) On random regular graphs of <i>N</i> = 1000 nodes and degree <i>k</i> = 4 with uniform epidemic parameters λ = 0.7, <i>μ</i> = 0.5. The partial observation is performed sampling uniformly the state of 30% of the nodes at <i>T</i>_<i>obs</i> = 4 and averaging over <i>M</i>_<i>o</i> = 150 such realizations.</p

Area under the ROC curve as function of the time <i>t</i> > <i>T</i>_<i>obs</i> = 3 on a Barabási-Albert random graph of <i>N</i> = 1000 nodes and average degree 〈<i>k</i>〉 ≈ 4 (with homogeneous epidemic parameters λ = 0.5, <i>μ</i> = 0.6), in the case of observation of a 30%-fraction of (a) nodes chosen at random uniformly and independently, (b) nodes forming a connected subgraph, (c) the most connected nodes.

<p>The average is computed over <i>M</i> = 201 epidemic realizations. The inference methods used are direct sampling with complete observation (black), random sampling (green), density sampling (blue), Similarity Sampling (magenta) and Belief Propagation (red).</p

Binary representation (<i>x</i>_1<i>a</i>, <i>x</i>_2<i>a</i>, <i>x</i>_3<i>a</i>) of the action space for the example presented in Section 1 with 3 users and 2 service units.

<p>Each vertex corresponds to a possible configuration: non-feasible configurations, that violate capacity constraints, are marked in red, Nash equilibria are marked in blue. The value of the aggregate utility is also reported for each configuration. The blue arrows indicate an improvement path obtained by best response from (0, 0, 1) to the Nash equilibrium (0, 1, 0).</p

Computation of the average utility U and of the entropy <i>S</i> as a function of the parameter <i>μ</i> for single instances with correlation <i>c</i> equal to −1, −0.5, 0, 0.5 and 1 (from top to bottom).

<p>The other parameters are <i>N</i> = 1 000, <i>M</i> = 100, <i>C</i> = 120, <i>q</i> = 0.2, <i>w</i>_min = 6, <i>w</i>_max = 15, <i>v</i>_min = 1 and <i>v</i>_max = 10. The values of </p><p><mo>U</mo></p> and <i>S</i> vs. <i>μ</i> are shown on the right-hand column of plots, together with the Bethe free energy <p></p><p><mi>μ</mi><mi>F</mi> <mo>=</mo> <mi>μ</mi><mo>U</mo> + <mi>S</mi></p><p></p>. The values of <p><mo>U</mo></p> and <i>μF</i> are read on the left-hand scale, the values of <i>S</i> on the right-hand one. For <i>c</i> ≤ −0.5 a first order transition, corresponding to a discontinuity in the first derivative of the free energy, is clearly visible. The critical value <i>μ</i>* is marked by a vertical grey line, and the thermodynamically unstable branches of the free energy are dotted. The entropy is plotted as a function of the average utility on the left-hand column of plots. For <i>c</i> ≤ −0.5 the entropy curve has two branches, separated by a wide gap (the high utility branches, which cover a small interval of utilities, are shown in detail in the insets, and they are highlighted by a black circle in the main plots). Dotted lines correspond to the thermodynamically unstable branches of the free energy. For <i>c</i> ≥ 0 the entropy curve is uninterrupted. The blue crosses at the top of each curve represent the values obtained with <i>μ</i> = 0, i.e. the uniform average over all Nash equilibria, and they always coincide with the maximum of the entropy. The histograms represent the distribution of the total utilities found by the dynamics G, BR and BRB. The number of runs for each dynamics is always 10⁵, except for BRB with <i>c</i> = −1 where it is 10⁶. The frequency of each value of utility can be read on the right-hand scale of the insets. The gray vertical lines are upper bounds to the total utility defined as <i>U</i>⁺ = ∑_<i>u</i> max_{<i>a</i> ∈ ∂<i>u</i>}<i>v</i>_<i>ua</i>.<p></p

Absolute value of the difference between the extinction time probability distribution <i>P</i>_<i>ext</i>(<i>t</i>) computed from direct sampling with complete information and those calculated with density sampling, BP and Similarity Sampling as a function of the number of infected and recovered nodes in the observed subset of nodes.

<p>a) On trees of <i>N</i> = 1092 nodes, with branching ratio 3 (〈<i>k</i>〉 ≈ 2) and with uniform epidemic parameters λ = 0.7, <i>μ</i> = 0.5. The partial observation is performed sampling uniformly the state of 10% of the nodes at <i>T</i>_<i>obs</i> = 5 and averaging over <i>M</i>_<i>o</i> = 210 such realizations. b) On random regular graphs of <i>N</i> = 1000 nodes and degree <i>k</i> = 4 and with uniform epidemic parameters λ = 0.7, <i>μ</i> = 0.5. The partial observation is performed sampling uniformly the state of 30% of the nodes at <i>T</i>_<i>obs</i> = 4 and averaging over <i>M</i>_<i>o</i> = 150 such realizations.</p

Average values U, D and C* of the observables as a function of the correlation <i>c</i> between weight and utility on individual edges.

<p>The other parameters are <i>N</i> = 1 000, <i>M</i> = 100, <i>C</i> = 120, <i>q</i> = 0.2, <i>w</i>_min = 6, <i>w</i>_max = 15, <i>v</i>_min = 1 and <i>v</i>_max = 10. Each data point is an average over 40 instances, and for each instance, each dynamics is realized 10 000 times. The average value of each observable is computed over the instances and over the realizations. The standard deviations (of the averages over the realizations of the dynamics across different instances) are much smaller than symbol sizes.</p

Predicted average epidemic size as function of the time <i>t</i> > <i>T</i>_<i>obs</i> = 3 on a Barabási-Albert random graph of <i>N</i> = 1000 nodes and average degree 〈<i>k</i>〉 ≈ 4 (with homogeneous epidemic parameters λ = 0.5, <i>μ</i> = 0.6), in the case of observation of a 30%-fraction of (a) nodes chosen at random uniformly and independently, (b) nodes forming a connected subgraph, (c) the most connected nodes.

<p>The average is computed over <i>M</i> = 201 epidemic realizations. The inference methods used are direct sampling with complete observation (black), density sampling (blue), Similarity Sampling (magenta) and Belief Propagation (red).</p

Average values U, D and C* of the observables as a function of the capacity <i>C</i>_<i>a</i> of service units and the correlation <i>c</i> between weight and utility on individual edges.

<p>The other parameters are <i>N</i> = 1 000, <i>M</i> = 200, <i>q</i> = 0.04, <i>w</i>_min = 6, <i>w</i>_max = 15, <i>v</i>_min = 1 and <i>v</i>_max = 10. Each data point, corresponding to a vertical line, is an average over 115 instances, and the standard deviations are of the order of the width of the lines.</p

Average aggregate utility of the initial configuration (black full line) and of the Nash equilibria found by best response (black circles) as a function of the parameter <i>γ</i> and for different values of the capacity <i>C</i> = 120 (panel A), 100 (B), 80 (C).

<p>The initial load of the units (red dashed line) and the load in the Nash equilibria (red squares) is also reported. The inset displays a magnified plot of the loads, showing that efficient Nash equilibria also induce a decrease in the load on the service units. In panel B we also report (on a different scale) the frequency of times we found good Nash equilibria in our numerical simulations.</p

Comparison of the minimum and maximum utilities found by Belief Propagation as <i>μ</i> → ± ∞ with the values found by Max-Sum, which allows to explicitly find an equilibrium configuration of extreme utility, and with the upper bound <i>U</i>⁺ defined in the main text.

<p>The last column shows the price of anarchy computed from the MS values.</p