Since I now know your question is theoretical, I can answer it better about how you can setup your network so you would be able to tell.  The exact way of setting it up will vary from device to device, but there are a lot of options available.  The key is that you need a device capable of logging that is in a location on the network that everything the person does will go through it.

For an average home network, this could be either the Wireless Access Point, the Router or if applicable, the gateway or firewall.  The wireless access point will get the information right as they get on to the network and will ensure that all activity coming in on the wireless network is logged, but if they were to use their wireless connection to compromise a wired system and then use the wired system, you would have an incomplete log.

Most likely the Router also serves as the gateway and firewall (the device that gives the rest of the network access to the Internet and ensures the Internet doesn't have unintended access to the private network).  Logging at the gateway and/or firewall ensures that all outbound traffic is logged, but won't tell you anything about what the intruder did internally on your network.

In most home networks, since there is only one router and it also generally serves as wireless access point, gateway and firewall, it will end up being privy to all connections flowing across the network.  It may not have access to all the contents of the packets it routes (if encryption is used) but it at least (by necessity) has access to the routing information (ie, where it is coming from and where it is going).  This is likely the best place in a home network to log.

Most consumer routers do not have this kind of logging by default, however third party firmwares such as DD-WRT often add logging capability to the routers that can either write to an internal store (limited space) or a network share.  Exactly what information can be stored and how it is formatted varies from device to device, but in general information like the IP and MAC Address of the client as well as the requested destination IP, Port and protocol (such as TCP or UDP) as well as a timestamp can give you a lot of what you need to get at least a general idea of what was going on.

If space is no object, it may even be possible to log the content of every packet sent across the network, but that will be a huge amount of space and it may not be safe to exclude "known" MAC addresses from the logging to try and reduce the amount since MAC spoofing is fairly easy on a wireless network.  (IE, the intruder can make themselves look like a valid machine)  Of course, if you are going to this kind of length, you could also do a per machine certificate and/or not use a weak key at which point the problem would likely be avoided in the first place.

How to Investigate Wi-Fi Intrusion and where to look for evidence?
Here is my JSFiddle.  What I'm trying to do is have a transition effect between two unordered lists.  When you click shop it takes you to the submenu.  When you click shop on  the submenu it takes you back to the main menu.  Why doesn't it slide or have any effect?  My goal is to transition the submenu from right to left, and have main menu transition left to right.  How can I make this work?

&lt;div class="mobile-nav"&gt;
    &lt;ul class="main-mobile-menu transition"&gt;
        &lt;li class="has-submenu"&gt;&lt;a href="#" title="SHOP" data-menu-id="shop"&gt;SHOP&lt;/a&gt;&lt;/li&gt;
        &lt;li class="has-submenu"&gt;&lt;a href="#" title="BRANDS" data-menu-id="brands"&gt;BRANDS&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href="#" title="ABOUT" &gt;ABOUT&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href="#" title="CONTACT" &gt;CONTACT&lt;/a&gt;&lt;/li&gt;
    &lt;/ul&gt;
    &lt;ul id="shop-submenu" class="hide transition"&gt;
        &lt;li&gt;&lt;a href="#" title="SHOP" data-menu-id="main-mobile-menu"&gt;SHOP&lt;/a&gt;&lt;/li&gt;
        &lt;li class="has-submenu"&gt;&lt;a href="#" title="ACCESSORIES" data-menu-id="accessories"&gt;ACCESSORIES&lt;/a&gt;&lt;/li&gt;
        &lt;li class="has-submenu"&gt;&lt;a href="#" title="ART" data-menu-id="art"&gt;ART&lt;/a&gt;&lt;/li&gt;
        &lt;li class="has-submenu"&gt;&lt;a href="#" title="WOMENS" data-menu-id="womens"&gt;WOMENS&lt;/a&gt;&lt;/li&gt;
        &lt;li class="has-submenu"&gt;&lt;a href="#" title="MENS" data-menu-id="mens"&gt;MENS&lt;/a&gt;&lt;/li&gt;
        &lt;li class="has-submenu"&gt;&lt;a href="#" title="SHOES" data-menu-id="shoes"&gt;SHOES&lt;/a&gt;&lt;/li&gt;
        &lt;li&gt;&lt;a href="#" title="GIFT CERTIFICATE"&gt;GIFT CERTIFICATE&lt;/a&gt;&lt;/li&gt;
    &lt;/ul&gt;
&lt;/div&gt;


This is my CSS:

.mobile-nav{
    position: absolute;
    height: 100%;
    padding: 0px;
    left: 0px;
    top: 0px;
    z-index: 0;
    overflow: hidden;
}

.transition{
    -webkit-transition: none 0.4s ease;
    -moz-transition: none 0.4s ease;
    -ms-transition: none 0.4s ease;
    -o-transition: none 0.4s ease;
    transition: none 0.4s ease;
    -webkit-transition-property: top, right, left, bottom, border;
    -moz-transition-property: top, right, left, bottom, border;
    -ms-transition-property: top, right, left, bottom, border;
    -o-transition-property: top, right, left, bottom, border;
    transition-property: top, right, left, bottom, border;
}

.hide{
    display: none;
}

.open{
    display: block;
}

