diff --git a/router/connection.go b/router/connection.go
index a42f04ce..a17a6fea 100644
--- a/router/connection.go
+++ b/router/connection.go
@@ -47,6 +47,7 @@ type LocalConnection struct {
 	stackFrag         bool
 	effectivePMTU     int
 	SessionKey        *[32]byte
+	heartbeatTCP      *time.Ticker
 	heartbeatTimeout  *time.Timer
 	heartbeatFrame    *ForwardedFrame
 	heartbeat         *time.Ticker
@@ -339,6 +340,7 @@ func (conn *LocalConnection) run(actionChan <-chan ConnectionAction, finished ch
 // goroutine, at least not without some synchronisation.
 
 func (conn *LocalConnection) initHeartbeats() error {
+	conn.heartbeatTCP = time.NewTicker(TCPHeartbeat)
 	conn.heartbeatTimeout = time.NewTimer(HeartbeatTimeout)
 	heartbeatFrameBytes := make([]byte, EthernetOverhead+8)
 	binary.BigEndian.PutUint64(heartbeatFrameBytes[EthernetOverhead:], conn.uid)
@@ -360,6 +362,8 @@ func (conn *LocalConnection) actorLoop(actionChan <-chan ConnectionAction) (err
 				break
 			}
 			err = action()
+		case <-conn.heartbeatTCP.C:
+			err = conn.sendSimpleProtocolMsg(ProtocolHeartbeat)
 		case <-conn.heartbeatTimeout.C:
 			err = fmt.Errorf("timed out waiting for UDP heartbeat")
 		case <-tickerChan(conn.heartbeat):
@@ -386,6 +390,7 @@ func (conn *LocalConnection) shutdown() {
 		conn.heartbeatTimeout.Stop()
 	}
 
+	stopTicker(conn.heartbeatTCP)
 	stopTicker(conn.heartbeat)
 	stopTicker(conn.fragTest)
 
@@ -439,6 +444,7 @@ func (conn *LocalConnection) receiveTCP(decoder *gob.Decoder) {
 
 func (conn *LocalConnection) handleProtocolMsg(tag ProtocolTag, payload []byte) error {
 	switch tag {
+	case ProtocolHeartbeat:
 	case ProtocolConnectionEstablished:
 		// We sent fast heartbeats to the remote peer, which has now
 		// received at least one of them and told us via this message.
@@ -469,7 +475,7 @@ func (conn *LocalConnection) handleProtocolMsg(tag ProtocolTag, payload []byte)
 }
 
 func (conn *LocalConnection) extendReadDeadline() {
-	conn.TCPConn.SetReadDeadline(time.Now().Add(ReadTimeout))
+	conn.TCPConn.SetReadDeadline(time.Now().Add(TCPHeartbeat * 2))
 }
 
 func (conn *LocalConnection) sendFastHeartbeats() error {
diff --git a/router/consts.go b/router/consts.go
index 4d8f312a..0aa13fd6 100644
--- a/router/consts.go
+++ b/router/consts.go
@@ -16,10 +16,10 @@ const (
 	UDPNonceSendAt      = 8192
 	FragTestSize        = 60001
 	PMTUDiscoverySize   = 60000
+	TCPHeartbeat        = 30 * time.Second
 	FastHeartbeat       = 500 * time.Millisecond
 	SlowHeartbeat       = 10 * time.Second
 	FragTestInterval    = 5 * time.Minute
-	ReadTimeout         = 1 * time.Minute
 	PMTUVerifyAttempts  = 8
 	PMTUVerifyTimeout   = 10 * time.Millisecond // gets doubled with every attempt
 	MaxDuration         = time.Duration(math.MaxInt64)
diff --git a/router/gossip.go b/router/gossip.go
index 96f4abe8..c0cfcde6 100644
--- a/router/gossip.go
+++ b/router/gossip.go
@@ -105,7 +105,7 @@ func (router *Router) NewGossip(channelName string, g Gossiper) Gossip {
 
 func (router *Router) SendAllGossip() {
 	for _, channel := range router.GossipChannels {
-		channel.Send(channel.gossiper.Gossip())
+		channel.Send(router.Ourself.Name, channel.gossiper.Gossip())
 	}
 }
 
@@ -167,7 +167,7 @@ func (c *GossipChannel) deliverBroadcast(srcName PeerName, _ []byte, dec *gob.De
 	return c.relayBroadcast(srcName, data)
 }
 
-func (c *GossipChannel) deliver(_ PeerName, _ []byte, dec *gob.Decoder) error {
+func (c *GossipChannel) deliver(srcName PeerName, _ []byte, dec *gob.Decoder) error {
 	var payload []byte
 	if err := dec.Decode(&payload); err != nil {
 		return err
@@ -175,26 +175,48 @@ func (c *GossipChannel) deliver(_ PeerName, _ []byte, dec *gob.Decoder) error {
 	if data, err := c.gossiper.OnGossip(payload); err != nil {
 		return err
 	} else if data != nil {
-		c.Send(data)
+		c.Send(srcName, data)
 	}
 	return nil
 }
 
-func (c *GossipChannel) Send(data GossipData) {
-	connections := c.ourself.Connections() // do this outside the lock so they don't nest
-	retainedSenders := make(connectionSenders)
+func (c *GossipChannel) Send(srcName PeerName, data GossipData) {
+	// do this outside the lock below so we avoid lock nesting
+	c.ourself.Router.Routes.EnsureRecalculated()
+	selectedConnections := make(ConnectionSet)
+	for name := range c.ourself.Router.Routes.RandomNeighbours(srcName) {
+		if conn, found := c.ourself.ConnectionTo(name); found {
+			selectedConnections[conn] = void
+		}
+	}
+	if len(selectedConnections) == 0 {
+		return
+	}
+	connections := c.ourself.Connections()
 	c.Lock()
 	defer c.Unlock()
-	for conn := range connections {
+	// GC - randomly (courtesy of go's map iterator) pick some
+	// existing entries and stop&remove them if the associated
+	// connection is no longer active.  We stop as soon as we
+	// encounter a valid entry; the idea being that when there is
+	// little or no garbage then this executes close to O(1)[1],
+	// whereas when there is lots of garbage we remove it quickly.
+	//
+	// [1] TODO Unfortunately, due to the desire to avoid nested
+	// locks, instead of simply invoking Peer.ConnectionTo(name)
+	// below, we have that Peer.Connections() invocation above. That
+	// is O(n_our_connections) at best.
+	for conn, sender := range c.senders {
+		if _, found := connections[conn]; !found {
+			delete(c.senders, conn)
+			sender.Stop()
+		} else {
+			break
+		}
+	}
+	for conn := range selectedConnections {
 		c.sendDown(conn, data)
-		retainedSenders[conn] = c.senders[conn]
-		delete(c.senders, conn)
 	}
-	// stop any senders for connections that are gone
-	for _, sender := range c.senders {
-		sender.Stop()
-	}
-	c.senders = retainedSenders
 }
 
 func (c *GossipChannel) SendDown(conn Connection, data GossipData) {
diff --git a/router/protocol.go b/router/protocol.go
index b9bb9f29..0fc548c6 100644
--- a/router/protocol.go
+++ b/router/protocol.go
@@ -8,7 +8,8 @@ const (
 type ProtocolTag byte
 
 const (
-	ProtocolConnectionEstablished ProtocolTag = iota
+	ProtocolHeartbeat ProtocolTag = iota
+	ProtocolConnectionEstablished
 	ProtocolFragmentationReceived
 	ProtocolStartFragmentationTest
 	ProtocolNonce
diff --git a/router/routes.go b/router/routes.go
index 3da3d1f2..1e854223 100644
--- a/router/routes.go
+++ b/router/routes.go
@@ -3,6 +3,7 @@ package router
 import (
 	"bytes"
 	"fmt"
+	"math"
 	"sync"
 )
 
@@ -77,6 +78,36 @@ func (routes *Routes) BroadcastAll(name PeerName) []PeerName {
 	return hops
 }
 
+// Choose min(log2(n_peers), n_neighbouring_peers) neighbours, with a
+// random distribution that is topology-sensitive, favouring
+// neighbours at the end of "bottleneck links". We determine the
+// latter based on the unicast routing table. If a neighbour appears
+// as the value more frequently than others - meaning that we reach a
+// higher proportion of peers via that neighbour than other neighbours
+// - then it is chosen with a higher probability.
+//
+// Note that we choose log2(n_peers) *neighbours*, not
+// peers. Consequently, on sparsely connected peers this function
+// returns a higher proportion of neighbours than elsewhere. In
+// extremis, on peers with fewer than log2(n_peers) neighbours, all
+// neighbours are returned.
+func (routes *Routes) RandomNeighbours(except PeerName) PeerNameSet {
+	res := make(PeerNameSet)
+	routes.RLock()
+	defer routes.RUnlock()
+	count := int(math.Log2(float64(len(routes.unicastAll))))
+	// depends on go's random map iteration
+	for _, dst := range routes.unicastAll {
+		if dst != UnknownPeerName && dst != except {
+			res[dst] = void
+			if len(res) >= count {
+				break
+			}
+		}
+	}
+	return res
+}
+
 func (routes *Routes) String() string {
 	var buf bytes.Buffer
 	routes.RLock()